7 Symmetry in the RelativisticMean Field Approximation

Joseph N. Ginocchio

MS B238, Los Alamos Nationonal Laboratory, Los Alamos, NM, 87545,gino@lanl.gov

Abstract. The Dirac Hamiltonian has an invariant SU(2) symmetry in two limits.For vector and scalar potentials that are equal in magnitude but opposite in sign, theDirac Hamiltonian is invariant under pseudo-spin symmetry. Approximate pseudo-spin symmetry in nuclei was observed in nuclear spectra more than thirty yearsago but its relativistic origin has only recently been discovered. The conditions onthe Dirac eigenfunctions imposed by pseudo-spin symmetry are derived and testedfor realistic relativistic mean field eigenfunctions. Predictions for magnetic momentsand Gamow-Teller transitions and nucleon-nucleus scattering are reviewed. Pseudo-spin symmetry is connected with quark degrees of freedom via a QCD sum rule.

For vector and scalar potentials that are equal, the Dirac Hamiltonian is invari-ant under spin symmetry. The possibility of approximate spin symmetry occurringfor an anti-nucleon in a nuclear enviroment is discussed. The exact eigenfunctionsand eigenenergies for the relativistic harmonic oscillator in this limit are derived.

7.1 Introduction

More than thirty years ago certain pairs of single-particle levels in sphericalnuclei were observed to be almost degenerate in energy [7.1,7.2]. The single-nucleon quantum numbers of these doublets are (nr, , j = + 1

2 ), (nr −1, + 2, j = − 1

2 ), where nr is the number of radial nodes, is the or-bital angular momentum, and j is the total angular momentum. For ex-ample, (1s 1

2, 0d 3

2), (2s 1

2, 1d 3

2), . . . , (1p 3

2, 0f 5

2), . . . , etc. Thus the states have

different radial quantum numbers as well as different orbital angular mo-menta. Introducing the pseudo-orbital angular momentum, the average ofthe orbital anular momentum = + 1, and the pseudo-spin s = 1

2 , thestates then were dubbed pseudo-doublets with j = ± 1

2 and the energyof the states in the doublet are then (approximately) independent of theorientation of the pseudo-spin. Pseudo-spin symmetry was also discoveredin the Nilsson model of rotational single-particle motion [7.3]. The orbits(N,n3, Λ,Ω = Λ + 1

2 ), (N,n3, Λ + 2, Ω = Λ − 12 ) are almost degenerate,

where N is the asymptotic total number of oscillator quanta and n3 is theasymptotic total number of oscillator quanta in the z direction, Λ is the or-bital angular mometum projected along the z direction, and Ω is the orbitalangular mometum projected along the z direction [7.4]. The pseudo-orbitalangular momentum projection would then be Λ = Λ + 1 and the energy of

J.N. Ginocchio, Symmetry in the Relativistic Mean Field Approximation, Lect. Notes Phys. 641,219–237 (2004)http://www.springerlink.com/ c© Springer-Verlag Berlin Heidelberg 2004

220 J.N. Ginocchio

(9/2(9/2-)

(7/2(7/2-)

5/25/2-

3/23/2-

1/21/2-

508.22508.22

333.26333.26

187.40187.40

74.3374.33

0

(11/2(11/2-)

(9/2(9/2-)

7/27/2-

5/25/2-

3/23/2-

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9.7469.746

[512]3/2[512]3/2[510]1/2[510]1/2

(kev)(kev) (kev)(kev)

Λ∼

= 1= 1

Fig. 7.1. The spectrum of two rotational bands built on pseudo-spin doublets in187Os

.

the doublets would be (approximately) independent of the orientation of thepseudo-spin Ω = Λ± 1

2 .

Pseudo-spin “symmetry” has been used to explain features of deformednuclei, including identical normal and superdeformed rotational bands [7.5–7.7] and alignment [7.8]. In Fig. 7.1 the spectrum of two bands in 187Osbased on the deformed orbits (5, 1, 0, Ω = 1

2 ), (5, 1, 2, Ω = 32 ), are shown as

an example. The bands are almost degenerate in energy. This fortitutousapproximate cancellation of the pseudospin-orbit single particle interaction[7.9,7.10] has been shown recently to result from a relativistic symmetry ofthe Dirac Hamiltonian that approximately occurs in nuclei because the vectormean field is approximately equal to the magnitude of the scalar mean field,but opposite in sign [7.11]. This relativistic symmetry is valid if the potentialsare spherical, axially deformed, or tri-axially deformed.

We shall discuss the symmetries of the Dirac Hamiltonian in Sect. 7.2. Re-alistic Dirac eigenfunctions are tested for pseudo-spin symmetry in Sect. 7.3.Pseudo-spin predictions for magnetic dipole and Gamow-Teller transtionsand for nucleon-nucleus scattering are reviewed in Sects. 7.4 and 7.5 respec-tively. The connection of pseudo-spin symmetry with QCD, the theory of thestrong interactions, is discussed in Sect. 7.6. In Sect. 7.7 we argue that spinsymmetry is expected for an anti-nucleon in a nuclear environment. Finallyin Sect. 7.8 we present the recently derived exact solution for the sphericalharmonic oscillator with spin symmetry which may be useful in the study ofthe anti-nucleon spectrum.

7 Symmetry in the Relativistic Mean Field Approximation 221

7.2 Symmetries of the Dirac Hamiltonian

The Lorentz covariant Dirac equation for a single particle with mass M is:[γµ[cpµ + gVAµ (xµ)] + M c2 + VS (xµ)

]Ψk(xµ) = 0, (7.1)

where xµ is the four spatial vector (ct, r), r is the three-dimensional vector(x, y, z), pµ = −i ∂

∂xµ, Aµ (xµ) is the Lorentz vector potential (A0 (xµ),

Ai(xµ)), i = 1,2,3, c is the speed of light, and VS (xµ) is the Lorentz scalarpotential. The Dirac matrices are four by four matrices

γ0 =(

1 00 −1

)= β, γi =

(0 σi−σi 0

)= β αi, i = 1, 2, 3 (7.2)

where 1 is the two dimensional unit matrix and σi are the two dimensionalPauli matrices. Assuming that the potentials are time independent, the Diracwavefunction can be factorized, Ψτ (xµ) = e−iEτ t

Φτ (r), and, after multiplyingthrough by β, the Dirac equation reduces to an eigenfunction equation,

H Φτ (r) = Eτ Φτ (r), (7.3)

whereH = α · [cp + gVA(r)] + VV (r) + β [Mc2 + VS(r)] (7.4)

is the Dirac Hamiltonian, α · [cp + gVA(r)] =∑3i=1 αi[cp + gVA(r)]i, and

VV (r) = gVA0(r) to conform to popular notation.In the mean field approximation the spatial components of the vector

potentia vanish, Ai(r) = 0, due to parity conservation. Using (7.2) the DiracHamiltonian in the matrix form then becomes

H =(

Mc2 + VV (r) + VS(r) σ · pσ · p −Mc2 − VV (r)− VS(r)

). (7.5)

Then the eigenstates of the Dirac Hamiltonian can be written as a four di-mensional vector,

Φτ (r) =

g+τ (r)

g−τ (r)

if+τ (r)

if−τ (r)

, (7.6)

where g±τ (r) are the “upper Dirac components” where + indicates spin up

and − spin down and f±τ (r) are the “lower Dirac components” where +

indicates spin up and − spin down.

7.2.1 Spin Symmetry

Spin symmetry occurs for VS(r) = V (r)2 + V 0

S , VV (r) = V (r)2 + V 0

V which isequivalent to VV (r)−VS(r) = Cs where Cs = V 0

V −V 0S is a constant. Although

222 J.N. Ginocchio

originally applied to the light mesons without success [7.12], recently spinsymmetry has been shown to be valid for mesons with one heavy quark forwhich the Dirac Hamiltonian is a valid approximation [7.13]. In Sect. 7.7 weargue that spin symmetry should also be valid for an anti-nucleon inside anucleus.

The Dirac Hamiltonian becomes

Hs =(

Mc2 + VV (r) + VS(r) σ · pσ · p −Mc2 + V 0

V − V 0S

). (7.7)

and the spin generators are [7.14]

Si =(

si 00 si

), (7.8)

where si = σi/2 and si = Up si Up where Up = σ·p|p| , |p| =

√p · p, is the helicity

unitary transformation, Up Up = 1 [7.10]. These generators form an SUs(2)algebra,

[Si, Sj ] = iεijk Sk (7.9)

and commute with the Dirac Hamiltonian (7.7)

[Si, Hs] =(

[si,Mc2 + VV (r) + VS(r)] siσ · p− σ · psisiσ · p− σ · psi [si,−Mc2 + V 0

V − V 0S ]

)= 0. (7.10)

The upper left entry is zero because the potentials do not depend on spin.The upper right and lower left entry are zero because Up σ ·p = σ ·p Up = |p|.These entries are zero for all Dirac Hamiltonians. The lower right entry iszero because −Mc2 + V 0

V − V 0S is a constant and thus commutes with the

momentum [pi,−Mc2 +V 0V −V 0

S ] = 0. Hence this entry is zero only for DiracHamiltionians with VV (r)− VS(r) = Cs.

Therefore this spin symmetry is an invariant symmetry of the DiracHamiltonian Hs. The eigenfunctions have spin 1

2 and spin projection, µ = ± 12 .

The eigenenergies are independent of the orientation of the spin,

Hs Φsk,µ(r) = Ek Φsk,µ(r), (7.11)

where k represents the non-spin quantum numbers and hence the eigenstateswill appear as spin doublets.

These doublets will be eigenfunctions of Sz with eigenvalue µ,

Sz Φsk,µ(r) = µ Φsk,µ(r). (7.12)

and the doublets will be connected by the spin raising and lowering operators,S±,

S± Φsk,µ(r) =

√(12∓ µ

)(32± µ) Φsk,µ±1(r). (7.13)

7 Symmetry in the Relativistic Mean Field Approximation 223

These conditions due to spin symmetry imply relationships between the am-plitudes of the eigenfunctions. Clearly, from the fact that the upper compo-nent of the spin generators (7.8) is simply si, (7.12) implies that [7.15]

g+k,− 1

2(r) = g−

k, 12(r) = 0, (7.14)

while (7.13) givesg+k, 12

(r) = g−k,− 1

2(r) = gk(r). (7.15)

Thus spin symmetry entails that the upper components of the spin partnershave the same spatial amplitude.

For the lower components the relationships are more complicated becausethe operator si intertwines spin and space due to the dependence on themomentum [7.15]:

f+k, 12

(r) = −f−k,− 1

2(r) = fk(r), (7.16)

(∂

∂x+ i

∂

∂y)f+k,− 1

2(r) = (

∂

∂x− i

∂

∂y)f−k, 12

(r), (7.17)

∂

∂zf±k,∓ 1

2(r) = ± (

∂

∂x∓ i

∂

∂y)f±k,± 1

2(r). (7.18)

Thus, the lower components (7.16) have the same spatial wavefunction butdiffer by a sign. However, the dominant lower components f±

k,∓ 12(r) can have

very different spatial wavefunctions.The Dirac wavefunctions in the doublet then become

Φsk, 12(r) =

gk(r)0

ifk(r)if−k, 12

(r)

, Φsk,− 12(r) =

0gk(r)

if+k,− 1

2(r)

−ifk(r)

. (7.19)

Thus, instead of eight amplitudes for the two states in the doublet, there arefour amplitudes, one upper and three lower, and the three lower are relatedby first order differential equations (7.17,7.18).

7.2.2 Pseudo-Spin Symmetry

Pseudo-spin symmetry occurs for VS(r) = V (r)2 + V 0

S , VV (r) = −V (r)2 + V 0

V

which is equivalent to VV (r)+VS(r) = Cps where Cps = V 0V +V 0

S is a constant[7.11]. The Dirac Hamiltonian becomes

Hps =(

Mc2 + V 0V + V 0

S σ · pσ · p −Mc2 + VV (r)− VS(r)

). (7.20)

and the pseudo-spin generators are [7.16]

224 J.N. Ginocchio

Si =(

si 00 si

), (7.21)

and these generators form an SUps(2) algebra,

[Si, Sj ] = iεijk Sk (7.22)

These generators commute with the Dirac Hamiltonian (7.20)

[Si, Hps] =(

[si,Mc2 + V 0V + V 0

S ] siσ · p− σ · psisiσ · p− σ · psi [si,−Mc2 + VV (r)− VS(r)]

)= 0.

(7.23)The upper right and lower right and left matrix elements are all zero for thesame reasons as for spin symmetry. In general, however, the upper left is zeroonly for the pseudo-spin limit, VV (r) + VS(r) = Cps = V 0

V + V 0S

Therefore this pseudo-spin symmetry is an invariant symmetry of theDirac Hamiltonian Hps. The eigenfunctions have pseudo-spin 1

2 and pseudo-spin projection, µ = ± 1

2 . The eigenenergies are independent of the orientationof the pseudo-spin,

Hps Φpsk,µ(r) = Ek Φpsk,µ(r), (7.24)

where k represents the non-pseudospin quantum numbers and hence theeigenstates will appear as pseudo-spin doublets.

These doublets will be eigenfunctions of Sz with eigenvalue µ,

Sz Φpsk,µ(r) = µ Φpsk,µ(r). (7.25)

and the doublets will be connected by the pseudo-spin raising and loweringoperators, S±,

S± Φpsk,µ(r) =

√(12∓ µ

)(32± µ) Φpsk,µ±1(r). (7.26)

These condtions due to pseudo-spin symmetry imply relationships betweenthe amplitudes of the eigenfunctions. Clearly, from the fact that the lowercomponent of the spin generators (7.21) is simply si, (7.25) implies that[7.11,7.17]

f+k,− 1

2(r) = f−

k, 12(r) = 0, (7.27)

while (7.26) givesf+k, 12

(r) = f−k,− 1

2(r) = fk(r). (7.28)

Thus pseudo-spin symmetry entails that the lower components of the pseudo-spin partners have the same spatial amplitude.

For the upper components the relationships are more complicated becausethe operator si intertwines spin and space due to the dependence on themomentum [7.15]:

7 Symmetry in the Relativistic Mean Field Approximation 225

g+k, 12

(r) = −g−k,− 1

2(r) = gk(r), (7.29)

(∂

∂x+ i

∂

∂y)g+k,− 1

2(r) = (

∂

∂x− i

∂

∂y)g−k, 12

(r), (7.30)

∂

∂zg±k,∓ 1

2(r) = ±(

∂

∂x∓ i

∂

∂y)g±k,± 1

2(r). (7.31)

Thus, the upper components in (7.29) have the same spatial wavefunctionbut differ by a sign. However, the dominant upper components, g±

k,∓ 12(r),

can have very different spatial amplitudes.Since the lower components are small compared to these upper compo-

nents, the upper components are observed experimentally. The reason it tookso long for the relativistic origin of pseudo-spin symmetry to be discoveredis because these upper components differ in their spatial amplitudes.

The Dirac eigenfunctions in the doublet then become

Φpsk, 12

(r) =

gk(r)g−k, 12

(r)ifk(r)

0

, Φpsk,− 1

2(r) =

g+k,− 1

2(r)

−gk(r)0

ifk(r))

. (7.32)

Thus, instead of eight amplitudes for the two states in the doublet, there arefour amplitudes, three upper and one lower, and the three upper are relatedby first order differential equations (7.30, 7.31).

7.3 Test for Pseudo-Spin Symmetry

indexpseudo-spin symmetry There are two types of relativistic theories thathave been used to describe relativistic dynamics in nuclei. The first is therelativistic theory of nucleons interacting by exchanging mesons, which hasa long history [7.18–7.20]. The other is a theory of relativistic nulceons inter-acting via point contact interactions [7.21]. Both of these relativistic theoriesare discussed in this volume. These theories have been solved in the rela-tivistic mean field approximation and, indeed, the resulting mean fields haveVV (r) ≈ −VS(r) which is consistent with the observation of approximatepseudo-spin symmetry in nuclei. The eigenfunctions from these calculationshave been tested to see if they satisfy the pseudo-spin symmetry conditions(7.27-7.31) and we shall review these tests [7.17,7.22,7.15,7.23].

7.3.1 Spherical Nuclei

For a spherical nuclei the mean field potentials depend only on the radialcoordinate, r =

√x2 + y2 + z2, and are independent of the polar angle, θ, z =

r cos(θ), and the azimuthal angle, φ, x = r sin(θ) cos(φ), y = r sin(θ) sin(φ).

226 J.N. Ginocchio

The Dirac Hamiltonian will be then be invariant with respect to rotationsabout all three axes, [Li, Hps] = 0 where

Li =(

i 00 i

), (7.33)

and hence invariant with respect to a SUL(2)×SUps(2) group where SUL(2)is generated by the orbital angular momentum operators Li. Since the totalangular momentum, Ji = Li + Si, is also conserved, rather than using thefour row basis for this eigenfunction, it is more convenient to introduce thespin function χµ explicitly. The states that are a degenerate doublet are thenthe states with j = ± 1

2 and they have the two row form [7.15]:

Ψpsnr,,j,M

(r) =

(gnr,,j

(r) [Y (j)(θ, φ) χ](j)Mifnr,

(r) [Y ()(θ, φ) χ](j)M

), (7.34)

where j = ± 1 for j = ± 12 , Y ()

m (θ, φ) is the spherical harmonic of order ,nr is the number of radial nodes of the lower amplitude, and [Y ()(θ, φ) χ](j)Mis the coupled amplitude

∑mµ C

12 j

mµMY()m (θ, φ) χµ. Thus the spherical sym-

metry reduces the number of amplitudes in the doublet even further fromfour to three. The Dirac eigenstates will then be an eigenfunction of L · L,J · J , and Jz,

J · J Ψpsnr,,j,M

(r) = j (j + 1) Ψpsnr,,j,M

(r), (7.35)

L · L Ψpsnr,,j,M

(r) = ( + 1) Ψpsnr,,j,M

(r), (7.36)

Jz Ψpsnr,,j,M

(r) = M Ψpsnr,,j,M

(r). (7.37)

The diferential relations (7.30,7.31) reduce to one equation in the sphericallimit,

(∂

∂r+

+ 2r

)gnr,,+ 12(r) = (

∂

∂r− − 1

r)gnr,,− 1

2(r). (7.38)

For pseudo-spin the radial amplitudes of the lower component in the dou-blet are equal. Therefore nr is the number of radial nodes of the lower ampli-tudes, not the upper amplitudes. The upper amplitude with j = − 1

2 in thedoublet has nr radial nodes while the upper amplitude with j = + 1

2 willhave nr - 1 radial nodes [7.24] which agrees with the pseudo-spin doubletsobserved in nuclei.

7.3.2 Test of Realistic Eigenfunctions with Spherical Symmetry

The lower components of the Dirac eigenfunctions for the pseudo-spin dou-blets using realistic eigenfunctions determined in relativistic mean field cal-culations have been shown to be approximately equal in a number of papers[7.17,7.25,7.26] and an example is shown in Fig. 7.2 for 2s 1

2, 1d 3

2(nr = 2, = 1).

7 Symmetry in the Relativistic Mean Field Approximation 227

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0 5 10 15

r (fm)

(fm )-3/ 2

2s

1d

f(r)

1/2

3/2

Fig. 7.2. The lower radial amplitude f(r) for the 2s 12

and 1d 32

eigenfunctions[7.17].

Recently the differential relations (7.38) satisfied by the upper compo-nents of the Dirac eigenfunctions in the pseudo-spin symmetry limit have beentested for the pseudo-spin doublets in spherical nuclei using realistic eigen-functions determined in relativistic mean field calculations [7.22,7.15,7.23].In Fig. 7.3a the upper components for the 1s 1

2and 0d3/2 eigenfunctions are

plotted (nr = 1, = 1); these eigenfunctions are very different in shape withdifferent numbers of radial nodes. In Fig. 7.3b the differential relations forthese eigenfunctions are plotted and we see a remarkable similarity betweenthe two differential relations except near the nuclear surface. In Fig. 7.3c theupper components for the 2s 1

2and 1d3/2 eigenfunctions are plotted (nr = 2,

= 1); likewise these eigenfunctions are very different in shape. In Fig. 7.3d thedifferential relations for these eigenfunctions are plotted and we see even bet-ter agreement between the two differential relations than for nr = 1. Similartests are made for higher radial quantum numbers and larger pseudo-orbitalangular momentum [7.15]. These results are for neutrons in 208Pb but similarconclusions hold for the protons as well. The pseudo-spin admixing decreasesfor increasing radial quantum number but decreasing pseudo-orbital angularmomentum, the same pattern followed by the binding energies [7.11] and thelower amplitudes of the eigenfunctions [7.17].

In the limit of small lower components, the upper components are thenon-relativistic approximation to the eigenfunctions. The differential rela-tions (7.38) have been tested as well for the non-relativistic eigenfunctions ofthe phemenological Woods-Saxon potential and self-consistent Hartree-Fockmean field [7.23]. The non-relativistic eigenfunctions are shown also to ap-proximately conserve pseudo-spin symmetry which is consistent with the factthat these models reproduce the single-nucleon spectrum well.

228 J.N. Ginocchio

-0.1

0

0.1

0.2

0.3

0.4

0.5

5 10 15-0.2

-0.15

-0.1

-0.05

0

0.05

0 5 10 15

a)

b)

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 5 10 15

r (fm)

-0.3

-0.2

-0.1

0

0.1

0.2

0 5 10 15

r (fm)

0 15

c) d)

g(r)

(fm) -3/2

g(r)

(fm) -3/2

Fig. 7.3. a) The upper component g(r) for the 1s 12

(solid line) and 0d 32

(dashedline) eigenfunctions, b) the differential equation on the right hand side (RHS) of(7.38) with = 1 for the 1s 1

2(solid line) eigenfunction and the differential equation

on the left hand side (LHS) of (7.38) with = 1 for the 0d 32

(dashed line) eigen-function, c) the upper component g(r) for the 2s 1

2(solid line) and 1d 3

2(dashed

line) eigenfunctions, and d) the differential equation on the RHS of (7.38) with =1 for the 2s 1

2eigenfunction (solid line) and the differential equation on the LHS of

(7.38) with = 1 for the 1d 12

(dashed line) eigenfunctions.

7.3.3 Pseudo-Spin Symmetry for Axially Deformed Nuclei

If the potentials are axially symmetric, that is, independent of the azimuthalangle φ, VS,V (r) = VS,V (ρ, z), ρ =

√x2 + y2, then the Dirac Hamiltonian has

an additional U(1) symmetry in the pseudo-spin limit. The Dirac Hamiltonianwill then be invariant under rotations about the z- axis, [Lz, Hps] = 0 where

7 Symmetry in the Relativistic Mean Field Approximation 229

Lz =(

z 00 z

), (7.39)

and z = Up zUp. Then the Dirac eigenstates are eigenstates of the U(1)generator, Lz, and the total angular momentum generator Jz = Sz + Lz,which is also conserved,

Lz ΦpsN,n3,Λ,µ

(r) = Λ ΦpsN,n3,Λ,µ

(r),

Jz ΦpsN,n3,Λ,µ,

(r) = Ω ΦpsN,n3,Λ,µ

(r), (7.40)

Ω = Λ + µ,

where the non-pseudospin quantum numbers of k of (7.32) are k = [N , n3, Λ],Λ is the pseudo - orbital angular momentum projection and Ω is the totalangular momentum projection onto the body-fixed z - axis. Although in thepseudo-spin limit the quantum numbers N , n3, Λ are the natural ones touse, the upper components, g∓

N,n3,Λ,± 12(ρ, z), are the dominate components

and are the ones identified experimentally. The non-relativistic asymptoticquantum numbers [7.4] are related to the pseudo-spin asymptotic quantumnumbers by [7.15]

N = N + 1, n3 = n3, Λ = Λ− 1. (7.41)

In the pseudo-spin limit and for deformed nuclei, the doublet eigenvectorshave the form [7.15]

ΦpsN,n3,Λ,

12(r) =

gN,n3,Λ(ρ, z) eiΛφ

g−N,n3,Λ,

12(ρ, z) ei(Λ+1)φ

ifN,n3,Λ(ρ, z) eiΛφ

0

, Ω′ = Λ +12, (7.42)

ΦpsN,n3,Λ,− 1

2(r) =

g+N,n3,Λ,− 1

2(ρ, z) ei(Λ−1)φ

−gN,n3,Λ(ρ, z) eiΛφ

0ifN,n3,Λ

(ρ, z) eiΛφ

, Ω = Λ− 12, (7.43)

that is,g+N,n3,Λ,

12(ρ, z) = −g−

N,n3,Λ,− 12(ρ, z) = gN,n3,Λ

(ρ, z), (7.44)

f+N,n3,Λ,− 1

2(ρ, z) = f−

N,n3,Λ,12(ρ, z) = 0, (7.45)

f+N,n3,Λ,

12(ρ, z) = f−

N,n3,Λ,− 12(ρ, z) = fN,n3,Λ

(ρ, z). (7.46)

For deformed nuclei the differential relations (7.30,7.31) become [7.15]

230 J.N. Ginocchio

(∂

∂ρ+

Λ + 1ρ

)g−N,n3,Λ,

12(ρ, z) = (

∂

∂ρ− Λ− 1

ρ)g+N,n3,Λ,− 1

2(ρ, z), (7.47)

∂

∂zg±N,n3,Λ,∓ 1

2(ρ, z) = ±(

∂

∂ρ± Λ

ρ)g∓N,n3,Λ,∓ 1

2(ρ, z). (7.48)

7.3.4 Test of Realistic Eigenfunctions with Axial Symmetry

As an example we show in Fig. 7.4 contour plots for the neutron pseudo-spinpartners [510]12 , [512] 32 ([411] 12 ,

32 ), in 168Er as a function of both ρ and z in

Fermis (fm) [7.27]. This Figure is broken down into three parts. In (A) therelationships between lower component amplitudes given in (7.45,7.46) areplotted. In the top row the amplitudes f±

k,∓ 12(r) are plotted, in the bottom

row f∓k,∓ 1

2(r) are plotted, and in the far right of the bottom row the differ-

ence between these two amplitudes is plotted to help assess how well theseamplitudes agree. In (B) the relationship between the upper component am-plitudes given in (7.44) are plotted. In the top row the amplitudes ∓g±

k,∓ 12

are plotted, in the bottom row ∓g∓k,∓ 1

2are plotted, and in the far right of the

bottom row the difference between these two amplitudes is plotted to helpassess how well these amplitudes agree. In (C) the differential relationshipsgiven in (7.47,7.48) are plotted. a and b are the left hand side (LHS) andright hand side (RHS) of (7.47), respectively, and the c is the difference be-tween them. d and e are the LHS and RHS of (7.48), respectively, and f isthe difference between them for Ω = Λ− 1/2. g and h are the LHS and RHSof (7.48), respectively, and i is the difference between them for Ω = Λ + 1/2.

From these figures, we can draw a number of conclusions. First of all, whilethe amplitudes f−

N,n3,Λ,− 12(ρ, z), f+

N,n3,Λ,12(ρ, z) are not zero as predicted

by (7.45) they are much smaller than f+N,n3,Λ,− 1

2(ρ, z), f−

N,n3,Λ,12(ρ, z). Fur-

thermore f+N,n3,Λ,− 1

2(ρ, z), f−

N,n3,Λ,12(ρ, z) have similar shapes and the same

number of nodes as predicted by (7.46). The amplitudes −g−N,n3,Λ,− 1

2(ρ, z)

have the same shape as g+N,n3,Λ,

12(ρ, z), as predicted by (7.44) but they dif-

fer in magnitude. These amplitudes are much smaller than the other upperamplitudes, which follows from spin symmetry [7.23]. Thus the amplitudesg±N,n3,Λ,∓ 1

2(ρ, z) are the dominant amplitudes and are very different shapes

which is the reason the origin of pseudo-spin symmetry eluded detection forthirty years.

The differential relation (7.47) between the dominant upper componentsis well obeyed in all cases. However the differential relations (7.48) relatingthe dominant upper components to the small upper components are not.

7 Symmetry in the Relativistic Mean Field Approximation 231

In both spherical and deformed nuclei the lower components are smallcompared to the upper components (Compare Fig. 7.2 with Fig. 7.3c andFig. 7.4A with the top row of Fig. 7.4B) which is consistent with the factthat nuclei are primarily non-relativistic quantum systems. However rela-tivistic quantum mechanics is necessary for the understanding of pseudo-spinsymmetry.

7.4 Magnetic Dipole and Gamow-Teller Transitions

Since the lower component of the Dirac wavefunction is small, the effect ofpseudo-spin symmetry on the relativistic wavefunction is difficult to test em-pirically except perhaps in certain forbidden transitions. For example, single -nucleon magnetic dipole and Gamow-Teller transitions between pseudo-spindoublets are forbidden non-relativistically because the orbital angular mo-menta of the two states differ by two units. However, they are not forbiddenrelativistically and therefore are sensitive to the lower component. As wehave seen in the last section, pseudo-spin symmetry predicts that the spa-tial amplitudes of the lower components of the Dirac eigenfunctions shouldbe equal and we have found that, indeed, they are approximately equal.The equality of the spatial amplitudes implies relationships between single-nucleon relativistic magnetic moments and magnetic dipole transitions withina pseudo-spin doublet, and between single-nucleon relativistic Gamow-Tellertransitions within a pseudo-spin doublet [7.28]. These relationships provide atest for the influence of pseudo-spin symmetry on the single - nucleon wave-functions.

For example, for neutrons, ν, if we know the magnetic moment, µj,ν , ofone of the partners, j = − 1

2 , we can predict the magnetic dipole transitionstrength between the two partners:

B(M1 : j′ = +12→ j = − 1

2)ν =

j + 12j + 1

[µj,ν − µA,ν ]2, (7.49)

where µA,ν is the analomous magnetic moment of the neutron. These re-lations have been tested for a range of nuclei and have been found to beapproximately valid [7.29]. The relations for Gamow-Teller transitions haveyet to be tested.

Not discussed here are the relations pseudo-spin symmetry predicts forquadrupole transitions between pseudo-spin doublets and the experimentalevidence supporting these relations [7.30].

7.5 Nucleon-Nucleus Scattering

The Dirac Hamiltonian is invariant with respect to pseudo-spin symmetryfor scalar and vector potentials equal and opposite in sign even if the po-tentials are complex. Indeed relativistic optical potentials describing nucleon

232 J.N. Ginocchio

Fig. 7.4. Contour plots of the [411] 12 , 3

2 pseudo-spin doublets eigenfunctions. Seethe text in Subsection 7.3.4 for details.

7 Symmetry in the Relativistic Mean Field Approximation 233

scattering from even-even nuclei approximately satisfy the pseudo-spin con-ditions [7.31]. Investigations show that for low nucleon energies pseudo-spinsymmetry is broken [7.32] but it improves as the energy increases [7.33–7.35].

7.6 QCD Sum Rules

QCD sum rules have been used to determine the nuclear scalar and vectormean fields in nuclear matter [7.36] and are reviewed in this volume. Usingaccepted values of the average quark mass, in the proton, mq ≈ 5 MeV, andthe sigma term, σN ≈ 45 MeV, the ratio of the scalar and vector fields is

VSVV

= − σN8 mq

≈ −1.125, (7.50)

which is uncannily close to the ratio of relativistic mean field potentials offinite nuclei at the origin, and indicative of pseudo-spin symmetry. The mi-nus sign comes from the fact that the quark condensate in the vacuum isnegative. These features suggest that perhaps pseudo-spin symmetry has amore fundamental foundation in terms of QCD.

7.7 Anti-nucleon Spectrum

Under charge conjugation the scalar potential remains invariant, VS(r) =C†VS(r)C = VS(r), but the vector potential changes sign, VV (r) =C†VV (r)C = −VV (r). Therefore for an anti-nucleon in a nuclear environ-ment VS(r) ≈ VV (r), and we have approximate spin symmetry [7.37]. In factthe negative energy solutions to the nucleon mean field do show a strong spinsymmetry [7.38]. However, there are self-consistent effects which mitigate thisconclusion [7.39]. Also the annihilation potential needs to be taken into ac-count to give a reliable prediction of the anti-nucleon spectrum [7.40]. But,since the annihilation potential exists only for the anti-nucleon mean fieldpotential and not the nucleon mean field potential, the annihilation potentialmust be equally scalar and vector so that it will vanish under charge conjuga-tion. This means that approximate spin symmetry will remain intact. Indeed,the limited polarized antinucleon scattering data available shows a vanishingsmall polarization which implies approximate spin symmetry [7.41].

7.8 Relativistic Harmonic Oscillatorwith Spin Symmetry

If the scalar and vector potentials are equal up to a constant, VS(r) = V (r)2 +

V 0S , VV (r) = V (r)

2 + V 0V , and are harmonic oscillator, V (r) = M

2 ω r2, withM = M + V 0

S (we have set = c = 1), the Dirac Hamiltonian is exactlysolvable even for the non-spherical harmonic oscillator [7.42].

234 J.N. Ginocchio

7.8.1 Eigenfunctions

The upper amplitudes of the eigenfunctions are

gnr,(r) = N (EN )

√2 λ3 n!

Γ ( + n + 32 )

e− x22 xL

(+ 12 )

nr (x2), (7.51)

where L(+ 1

2 )n (x2) is the Laguerre polynomial, x = λr, λ =

[(EN + M) Mω2

2

] 14,

and

N (E) =

√2 (E + M)3E + M

, (7.52)

with E = E − V 0V . The lower components are

fnr,,j=− 12(r) = (7.53)

− N (EN )M + EN

√2 λ5 n!

Γ ( + nr + 32 )

e− x22 x−1((nr + 1) L

(− 12 )

nr+1 (x2) + (nr + +12) L

(− 12 )

nr (x2)),

(7.54)

fnr,,j=+ 12(r) =

N (EN )M + EN

√2 λ5 n!

Γ ( + nr + 32 )

e− x22 x+1(L(+ 3

2 )nr (x2) + L

(+ 32 )

nr−1 (x2))

(7.55)The function fnr,,j=− 1

2(r) has nr + 1 nodes, one more node than the upper

component. The amplitude fnr,,j=+ 12(r) has the same number of nodes as

the upper component. This agrees with the general theorem relating thenumber of radial nodes of the lower comonents to the number of radial nodesof the upper component [7.24].

7.8.2 Energy Eigenvalues

There are three eigenvalue solutions:

E(1)N = M

[B(AN ) +

13

+4

9 B(AN )

]+ V 0

V , (7.56)

E(2)N = M

[− (1− i

√3)

2B(AN ) +

13− 2 (1 + i

√3)

9 B(AN )

]+ V 0

V , (7.57)

E(3)N = M

[− (1 + i

√3)

2B(AN ) +

13− 2 (1− i

√3)

9 B(AN )

]+ V 0

V , (7.58)

where

B(AN ) =

[27A2

N + 3√

3√

27A2N − 32 AN − 16

54

] 13

, (7.59)

7 Symmetry in the Relativistic Mean Field Approximation 235

AN = C (N + 32 ), C =

√2 ωM

, and N is the total oscillator quantum number,N = 2n + = 0, 1, . . .. We note that there is not only a degeneracy due tospin symmetry but there is also the usual degeneracy of the non-relativisticharmonic oscillator; namely, that the energy depends only on the total har-monic oscillator quantum number and the states with = N,N − 2, . . . , 0 or1 are all degenerate.

The eigenvalues E(1)N are real for all values of N as long as C, V 0

S,V are real

[7.42]. The eigenvalues E(2,3)N are also real for 0 ≤ AN ≤

√3227 . However for

AN >√

3227 , E

(2,3)N are complex and the complex conjugates of one another.

For AN = 0, E(2)N = −M + VV while E

(1,3)N = M + V 0

V . Possibly E(2)N are the

“negative” energy eigenvalues, but this does not seem compelling because,as the harmonic oscillator strength increases, more of the spectrum becomes

complex so that for C > 23

√3227 all of the values of E

(2,3)N are complex. The

solutions E(2,3)N seem to be a peculiarity of the relativistic harmonic oscilator.

Therefore we assume that the bound spectrum of the relativistic harmonicoscillator is given by E

(1)N .

The spectrum is non-linear in contrast to the non-relativistic harmonicoscillator. However for small AN

E(1)N ≈ M (1 +

AN√2

+ · · ·) + V 0V (7.60)

and therefore the binding energy, E(1)N − M ≈ ω (N + 3

2 ) + V 0V , in agreement

with the non-relativistic harmonic oscillator. For large AN the spectrum goesas

E(1)N ≈ M (A

23N + · · ·) + V 0

V , (7.61)

which, in lowest order, agrees with the spectrum for M → 0 [7.43].

7.8.3 Pseudo-Spin Symmetry

As discussed in Sect. 7.3, nucleons in a nuclear mean field exhibit approxi-mate pseudo-spin symmetry [7.11]. The harmonic oscillator can be solved forthis system also. The eigenenergies, E

(i)k , are the negative of the eigenener-

gies for spin symmetry since the antinucleon and nucleon systems are chargeconjugates of each other [7.37]; that is, E

(i)k = −E

(i)k . Hence the eigenfunc-

tions with eigenenergies E(1)k correspond to the negative energy states of

the Dirac Hamiltonian, the sea states, and the psuedo-quantum numbers arethen the quantum numbers of the lower amplitudes [7.11,7.24]. Possibly thevalence Dirac states with positive energy are those with eigenenergies E

(2)k .

This needs to be investigated.

236 J.N. Ginocchio

7.9 Future - Beyond the Mean Field

We have reviewed in this paper the evidence that pseudo-spin is approxi-mately conserved in the relativistic and non-relativistic mean field approx-imation. The fact that QCD sum rules support approximate pseudo-spinsymmetry suggests a more fundamental rationale for pseudo-spin symme-try. Does the nucleon-nucleon interaction conserves pseudo-spin symmetry?An investigation of the nucleon-nucleon scattering matrix has shown thatthe pseudo-spin symmetry generated by the pseudo-spin operators in (7.21)does not conserve pseudo-spin symmetry [7.44]. However, these generatorsare appropriate for a mean field approximation of relativistic field theory. Inthis appproximation the spatial components of the vector field, Ai(r), arezero. However, there is a generalized pseudo-spin symmetry for Ai(r) = 0[7.14]. The nucleon-nucleon scattering matrix is presently being investigatedto determine if it conserves this generalized pseudo-spin symmetry.

This work was supported by the U.S. Department of Energy under con-tract W-7405-ENG-36.

References

[7.1] A. Arima, M. Harvey and K. Shimizu: Phys. Lett. B30, 517 (1969).[7.2] K.T. Hecht and A. Adler: Nucl. Phys. A137, 129 (1969).[7.3] A. Bohr, I. Hamamoto, and B. R. Mottelson: Phys. Scr. 26, 267 (1982).[7.4] A. Bohr and B. R. Mottelson: Nuclear Structure, Vol. II (W. A. Benjamin,

Reading, Ma., 1975).[7.5] W. Nazarewicz, P.J. Twin, P. Fallon and J.D. Garrett: Phys. Rev. Lett. 64,

1654 (1990).[7.6] F.S. Stephens et al.: Phys. Rev. Lett. 65 301 (1990); F.S. Stephens et al.:

Phys. Rev. C 57 R1565 (1998).[7.7] A.M. Bruce et. al.: Phys. Rev. C 56, 1438 (1997).[7.8] B. Mottelson: Nucl. Phys. A522, 1 (1991).[7.9] C. Bahri, J. P. Draayer, and S. A. Moszkowski: Phys. Rev. Lett. 68, 2133

(1992).[7.10] A. L. Blokhin, C. Bahri and J. P. Draayer: Phys. Rev. Lett. 74, 4149 (1995).[7.11] J.N. Ginocchio: Phys. Rev. Lett. 78, 436 (1997).[7.12] G. B. Smith and L. J. Tassie: Ann. Phys. 65, 352 (1971).[7.13] P. R. Page, T. Goldman, and J.N. Ginocchio: Phys. Rev. Lett. 86, 204

(2001).[7.14] J. S. Bell and H. Ruegg: Nucl. Phys. B98, 151 (1975).[7.15] J.N. Ginocchio: Phys. Rev. C66, 064312 (2002).[7.16] J.N. Ginocchio and A. Leviatan: Phys. Lett. B425, 1 (1998).[7.17] J.N. Ginocchio and D. G. Madland: Phys. Rev. C57 1167 (1998).[7.18] B. D. Serot and J. D. Walecka: The Relativistic Nuclear Many - Body

Problem in Advances in Nuclear Physics, edited by J. W. Negele and E.Vogt, Vol. 16 (Plenum, New York, 1986).

[7.19] B. D. Serot, J. D. Walecka: Int.J.Mod.Phys. E6, 515 (1997).

7 Symmetry in the Relativistic Mean Field Approximation 237

[7.20] P. Ring: Prog. Part. Nucl. Phys. 37, 193 (1996).[7.21] B. A. Nikolaaus, T. Hoch and D. G. Madland: Phys. Rev.C46, 1757 (1992).[7.22] J.N. Ginocchio and A. Leviatan: Phys. Rev. Lett. 87, 072502 (2001).[7.23] P.J. Borycki, J. Ginocchio, W. Nazarewicz, and M. Stoitsov: Phys. Rev.

C68, 014304 (2003).[7.24] A. Leviatan and J.N. Ginocchio: Phys. Lett. B518, 214 (2001).[7.25] G. A. Lalazissis, Y. K. Gambhir, J. P. Maharana, C. S. Warke and P. Ring:

C58, R45 (1998).[7.26] J. Meng, K. Sugawara-Tanabe, S. Yamaji, P. Ring and A. Arima: Phys. Rev.

C58, R628 (1998).[7.27] J.N. Ginocchio, A. Leviatan, J. Meng, and S.-Z. Zhong: to be published in

Phys. Rev. C (2003).[7.28] J. N. Ginocchio: Phys. Rev. C59, 2487 (1999).[7.29] P. von Neumann-Cosel and J. N.Ginocchio: Phys. Rev. C62, 014308 (2000).[7.30] J.N. Ginocchio: to be published in Phys. Rep. (2004).[7.31] E. D. Cooper, S. Hama, B. C. Clark, and R. L. Mercer: Phys. Rev. C47,

297 (1993).[7.32] J. B. Bowlin, A. S. Goldhaber and C. Wilkin: Z. Phys. A331, 83 (1988).[7.33] J. N. Ginocchio: Phys. Rev. Lett. 82, 4599 (1999).[7.34] H. Leeb and S. Wilmsen: Phys. Rev. C62, 024602 (2000).[7.35] H. Leeb and S. A. Sofianos: to be published in Phys. Rev. C (2003); Elec-

tronic Archives: nucl-th/0304009.[7.36] T. D. Cohen, R. J. Furnstahl, K. Griegel, and X. Jin: Prog. in Part. and

Nucl. Phys. 35, 221 (1995).[7.37] J.N. Ginocchio: Phys. Rep. 315, 231 (1999).[7.38] S. G. Zhou, J. Meng, and P. Ring: Phys. Rev. Lett. (2003) in print; Nucl-

th/0304067.[7.39] T. Burvenich et al : Phys. Lett. B542, 261 (2002).[7.40] T. Burvenich and J. N. Ginocchio: work in progress (2003).[7.41] D. Garetta et al , Phys. Lett. B151, 473 (2002).[7.42] J. N. Ginocchio: submitted to Phys. Rev. C (2003).[7.43] R. K. Bhadri: Models of the Nucleon: From quarks to Soliton, (Addison-

Wesley, 1988).[7.44] J.N. Ginocchio: Phys. Rev. C65, 054002 (2002).